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Notebook[{

Cell[CellGroupData[{
Cell["Unusual sequence stuff", "Subsubtitle"],

Cell[TextData[{
  "Create a sequence by starting with two given fixed numbers ",
  Cell[BoxData[
      \(TraditionalForm\`a\_0\)]],
  "and ",
  Cell[BoxData[
      \(TraditionalForm\`a\_1\)]],
  "and create a new sequence by appending the digits of their product to the \
end.  Continuing in that fashion one step to the right, etc..."
}], "Text"],

Cell[BoxData[
    \(seq[a_, b_, 
        steps_] := \(Nest[\(({Join[#[\([1]\)], 
                  Flatten[
                    Table[IntegerDigits[\(#[\([\)\(1\)\(]\)]\)[\([i]\)]*\(#[\(\
[\)\(1\)\(]\)]\)[\([i + 1]\)]], \ {i, \ #[\([2]\)], \ 
                        Length[#[\([\)\(1\)\(]\)]] - 1}]]], 
                Length[#[\([1]\)]]})\) &, {{a, b}, 1}, 
          steps]\)[\([1]\)]\)], "Input"],

Cell["\<\
Here is a sample output.  Example taken from \"Problem Solving \
through Problems by Larson:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(seq[2, \ 7, \ 6]\)], "Input"],

Cell[BoxData[
    \({2, 7, 1, 4, 7, 4, 2, 8, 2, 8, 8, 1, 6, 1, 6, 1, 6, 6, 4, 8, 6, 6, 6, 
      6, 6, 3, 6, 2, 4, 3, 2, 4, 8, 3, 6, 3, 6, 3, 6, 3, 6}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "We are asked to prove that the digit 6 appears an infinite number of times \
in this sequence.\n\nI did this by noticing the following:\n\nLet ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(<\)\(S\_1\)\(>\)\) \[RightArrow] \(\(<\)\(S\_2\)\
\(>\)\)\)]],
  "mean that sequence ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(S\_1\)\(\ \)\)\)]],
  "appends ",
  Cell[BoxData[
      \(TraditionalForm\`S\_2\)]],
  " to the end of the sequence.  Notice that ",
  Cell[BoxData[
      \(TraditionalForm\`S\_2\)]],
  "stays in tact and when we reach it, it will append its portion to the end. \
 This reduces the problem to finding a finite sequence that guaruntees \
reproduction of 6s.\n\nLooking above at the output we find <3, 6, 3> and have \
the following:\n",
  Cell[BoxData[
      \(TraditionalForm\`\(\(<\)\(3\)\), 
      6, \(\(3\)\(>\)\) \[RightArrow] \(\(<\)\(1\)\), \ 8, \ 
      1, \ \(\(8\)\(>\)\)\  \[RightArrow] \ \(\(<\)\(8\)\), \ 
      8, \ \(\(8\)\(>\)\)\  \[RightArrow] \ \(\(<\)\(6\)\), \ 4, \ 
      6, \ \(\(4\)\(>\)\)\  \[RightArrow] \ \(\(<\)\(2\)\), \ 4, \ 2, \ 4, \ 
      2, \ \(\(4\)\(>\)\)\  \[RightArrow] \ \(\(<\)\(8\)\), \ 8, \ 
      8 > \(+etc\) ... \)], "InlineFormula"],
  "\n\nEventually we run into another string of three 8's (a long string in \
fact) which is immediately followed by appending <6, 4, 6, 4>.  This repeats \
ad infinitum for an infinite number of 6's."
}], "Text"]
}, Open  ]]
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(*******************************************************************
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Cell[CellGroupData[{
Cell[1776, 53, 45, 0, 33, "Subsubtitle"],
Cell[1824, 55, 347, 9, 50, "Text"],
Cell[2174, 66, 400, 8, 91, "Input"],
Cell[2577, 76, 116, 3, 32, "Text"],

Cell[CellGroupData[{
Cell[2718, 83, 49, 1, 27, "Input"],
Cell[2770, 86, 164, 2, 43, "Output"]
}, Open  ]],
Cell[2949, 91, 1435, 30, 244, "Text"]
}, Open  ]]
}
]
*)



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End of Mathematica Notebook file.
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